Have your county agent look at the beds & plants, do a soil test.

Start by contacting you counties office of the Alabama Cooperative Extension service (supported by Alabama A & M and Auburn Universities) about having a good, reliable soil test to find out the pH of that soil and the amounts of the major nutrients in that soil. You might also want to use these simple soil tests,
1) Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains� too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.
to see what you have now and help you determine what you may need to do.
Your plants could be having problems with nutrients because 1. they are not available, 2. nutrients are not available because of the soils pH, 3. the Soil Food Web is not yet working and feeding your plants.
Spraying pesticides without identifying the pest problem is not organic. Keep in mind that Neem products are pretty broad spectrum poison just an most other organic pesticides and the synthetic ones.

I'll only address one thing: yes, I think mulch is pretty much vital and if you don't mulch, weeds tend to go crazy while desired plants suffer in the scorched, overheated soil.
Ok, I'm being a bit dramatic, but only a bit!

Soil test sent off today - hopefully I will hear something soon! Things are looking minimally better, so I am hopeful a big part of the problem was overwatering.

Kimmsr: If I can't see the pests, how do I identify what is eating them? The Neem came highly recommended, but I have never used pesticides before. This is all confusing and new for me as I have always had excellent luck with my flower gardens and general landscaping over the past 10 years. No bugs and very few stressed plants. Vegetables and organic gardening are going to be a learning curve, but I feel it will be well worth it!

feijoas: What would you suggest I mulch with? I read about being careful when adding things that haven't properly broken down yet (might steal N from the soil), so would I use something bagged from a nursery? I'm not worried about weeds as the beds are raised and no weeds appearing so far. The soil is getting pretty hard on the top layer though, so I would like to do something.

Thanks everyone!

Do the plants have small holes in them indicating they are being eaten by insects. Are the plants chomped on by larger animals such as rabbits? Are there irregular shaped holes indicating maybe slugs and/or snails? All of the pests leave clues abnout who they are, we just need to look at those clues and properly interpret them.
Mulches are materials we put down on the surface of the soil to 1) help suppress unwanted plant growth (weeds), 2) aid in soil moisture conservation, 3) aid in keeping the soil cool, and 4) add organic matter to our soil. Some people, not understanding those reasons, often will say putting raw, undigested materials on soil could cause the soil bacteria to divert to digesting that material and tieing up Nitrogen in the process. That does not happen. Those people that do not understand work that raw, undigested material into the soil, a soil amendment now not a mulch, where the soil bacteria will get busy and use Nitrogen your plants need, although that N will be put back into the soil as the bacteria die. On the soil no Nitrogen is diverted, in the soil plants could not have enough Nitrogen to grow.

kimmsr: Thank you very much!! I've always mulched around my flowers and shrubs and never really thought about the chemistry behind it. Then I became alarmed reading about N depletion, but you have cleared that up for me. So, shredded leaves would be an acceptable surface mulch? Other suggestions?

As to pest damage, the edges of the cucumber leaves look torn and ragged. I know we don't have rabbits here (I live in an urban area), only opossums, squirrels and chipmunks that I have seen over the past 7 years. The peppers have small round holes mostly in the middle of the leaves. I am going to search out some gardening books that may help in pest identification. I did notice some very tiny bugs under the soil in a different area of the bed. I also have noticed ants...ugh. I'm trying cornmeal for the ants and I've planted mint around which I read deters them. (I understand I will have to keep a very close eye on the mint as it's invasive).

Thanks again!

If you have Opossums, Squirrels, and Chipmonks you also have Rabbits, they do live in urban environments, maybe groundhogs even. Slugs, Snails, several of the beetles come to mind. Not all leaf chewers stay around for you to identify, easily, but spraying some broad spectrum poison to control something also will kill off the beneficial insects, many of which are predators of the pests. Not all "bugs" should be eliminated since many are part of Ma Natures recycling machine and help get rid of our waste and convert that waste into nutrients our plants can use. That does include ants. Cornmeal will do nothing to help control ants. I know ants will avoid Peppermint Oil but I see ants on the spearmint I have growing here.
Companion planting can help deter insect pests, but what to plant depends on the insect pest you want to keep away.

Here is a link that might be useful: Companion planting

On my vege gardens I use a pretty random mix of dried lawn cippings (basically no systemics used on grass over here), cut weeds, old straw, soft perennial prunings, seaweed, etc, etc.
As kimmsr says, nitrogen tie-up is only an issue if a high-carbon mulch like bark is actually dug into the soil.
I use a LOT of chipped tree mulch, but only in areas I'm not going to disturb, like around perennials.